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Chapter8_IntervalEstimation

# Chapter8_IntervalEstimation - Chapter 8 Interval Estimation...

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Chapter 8 Interval Estimation

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We have talked about… Parameters μ , σ 2 , p Statistics Sample mean, s 2 , and sample proportion Sampling distributions Reliability We want to make inferences about μ , σ 2 , and p
Statistical Inferences Estimation When we want to estimate or predict the value of a population parameter Statistical Tests Conduct a test to determine if the value of the parameter is equal to some pre-specified or hypothesized value

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Types of Estimates Point Estimate Single number calculated from the data that is used to estimate the parameter. This is the statistic . Interval Estimate Involves the calculation of two numbers from the sample to provide bounds for the parameter. Computed by adding and subtracting a margin of error to the point estimate Why do we need an interval estimate?
Estimation for μ , σ Known . x of on distributi sampling the of properties the using by found is for estimate interval The . x mean, sample the calculate and population the from n size of sample a collect we , estimate To mean. sample the , x is for estimate point The μ μ μ

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Estimation for μ , σ Known (continued) CLT: When n is large enough (n ≥ 30), the sampling distribution of the sample mean is approximately normally distributed with a mean of μ and a variance of σ 2 /n. The interval estimate we will develop is called the Confidence Interval (CI)
Estimation for μ , σ Known (continued) In order to develop an interval estimate of a population mean, the margin of error must be computed using either: the population standard deviation σ , or the sample standard deviation s 2200 σ is rarely known but a good estimate can be obtained based on historical information. This is the σ -known case.

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Estimation for μ , σ Known (continued) Interval estimate is not 100% certain to contain μ But we want the probability of making an error to be as small as possible Let α = probability that the confidence interval does not contain μ We want α to be “small.” Typical values: 0.01, 0.02, 0.05, and 0.10
Estimation for μ , σ Known (continued) We want to find a 100(1 – α )% CI for μ . (1- α ) is called the confidence coefficient If α = 0.05, then we want a 100(1–0.05)% = 95% CI for μ

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μ α /2 α /2 1 -  α   of all      values x Sampling distribution of   x x z x α σ /2 z x α σ /2 Estimation for μ , σ Known (continued)
Estimation for μ , σ Known (continued) We need to find the values of z α /2 such that P(0 < Z < z α /2 ) = (1 – α )/2 The bounds on the 100(1 – α )% CI for μ is n σ ± α /2 z x

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Estimation for μ , σ Known (continued) When σ (or σ 2 ) is unknown and n is large (n ≥ 30), the value of σ can be estimated by s (s 2 ), the sample standard deviation.
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Chapter8_IntervalEstimation - Chapter 8 Interval Estimation...

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